Qubit purification speed-up for three complementary continuous measurements

0.25 MB·English

by  Rusko Ruskov

Qubit purification speed-up for three complementary continuous measurements
Rusko Ruskov,1,2 Joshua Combes,3,4 Klaus Mølmer,1 and Howard M. Wiseman3
1Lundbeck Foundation Theoretical Center for Quantum System Research,
Department of Physics and Astronomy, ˚Arhus University, DK-8000 ˚Arhus C, Denmark
2Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, MD 20740, USA
3 Centre for Quantum Computation and Communication Technology (Australian Research Council),
2
1 Centre for Quantum Dynamics, Griffith University, Brisbane, 4111, Australia
0
4Center for Quantum Information and Control,
2

n University of New Mexico, Albuquerque, NM 87131-0001, USA
a
J (Dated: January 19, 2012)

8
1 We consider qubit purification under simultaneous continuous measurement of the three

] non-commutingqubitoperatorsσˆx,σˆy,σˆz. Thepurificationdynamicsisquantifiedby(i)the
h
p averagepurificationrate,and(ii)themeantimeofreachinggivenlevelofpurity,1 ε. Under
- −
t idealmeasurements(detectorefficiencyη =1),weshowinthefirstcaseanasymptoticmean
n
a
purificationspeed-upof4ascomparedtoastandard(classical)single-detectormeasurement.
u
q However by the second measure — the mean time of first passage T¯(ε) of the purity — the
[

correspondingspeed-upisonly2. Weexplainthesespeed-upsusingtheisotropyofthequbit
1
v evolutionthatprovidesanequivalencebetweentheoriginalmeasurementdirectionsandthree
2
9 simultaneous measurements, one with an axis aligned along the Bloch vector and the other
8
3 with axes in the two complementary directions. For inefficient detectors, η = 1 δ < 1
−
.
1 the mean time of first passage T¯(δ,ε) increases since qubit purification competes with an
0
2 isotropic qubit dephasing. In the asymptotic high-purity limit (ε,δ 1) we show that
≪
1
: the increase possesses a scaling behavior: ∆T¯(δ,ε) is a function only of the ratio δ/ε. The
v
i increase ∆T¯(δ/ε) is linear for small argumentbut becomes exponential exp(δ/2ε) for δ/ε
X ∼
r large.
a
PACS numbers: quantummeasurement, quantumcontrol, quantum feedback, qubit,purification
I. INTRODUCTION
Pure states are an important resource in quantum computation and communication algorithms
[1, 2]. While state purification is possible via cooling, this may be impractical for several reasons,
including long relaxation times, degenerate ground states, and the presence of several dephas-
ing mechanisms. A different purification process is possible via continuous measurement when
the (available to the observer) quantum state will purify continuously according to the detector
2
measurement result. Here, the speed at which one can purify the state is set by the detector mea-
surement rate, and the final purity will depend on how close the detector is to a quantum-limited
(100% efficient) one. This may become an important tool since continuous measurements are also
at the heart of various quantum control applications[2], including quantum state stabilization via
quantum feedback [3–5], preparation of entangled states [6], and for continuous error corrections
[7].
In recent years several groups have suggested rapid purification protocols based on continuous
measurement and Hamiltonian feedback [8–12], which makes it possible to considerably speed-up
purification. For a single qubit, the problem was first analyzed by Jacobs [8], who recognized that
therateof average purificationcanbeenhancedusingunitarytransformation ateach measurement
time step, so as to make the state always orthogonal (in the Bloch sphere sense) to the detector’s
measurement basis. This feedback algorithm (which has been rigorously shown to be optimal [13],
and rederived in Ref. [14]) produces a factor of 2 speed-up in the high-purity limit, when the
state approaches the Bloch sphere surface. The speed-up is by comparison with a no-feedback
measurement, which can be completely understood classically (see below). This speed-up is a
quantum mechanical effect since it is only possible if the system can exist in superposition of the
eigenstates of the measured observable.
Jacobs’ protocol maximizes the average purification rate in the sense that it minimizes the
time τ it takes the average purity p to reach a certain level 1 ε. A different, but equally well
h i −
motivated, aim was suggested by Wiseman and Ralph [10], that of minimizing the mean time T
h i
for the purity p to attain a certain level. It was argued (and later rigorously confirmed [13, 14])
that the optimal protocol in this case is to rotate the state so that at each measurement time step
it is aligned with the detector (i.e., making the state density matrix diagonal in the basis of the
detector observable). This is the classical protocol referred to above, and in the absence of any
other dynamics, requires no feedback to realize.
In this paper we study a recently suggested purification protocol based on monitoring of the
qubit state via simultaneous continuous measurement of the three non-commuting qubit operators
[15]σˆ ,σˆ ,σˆ (Fig.1). (Arelatedprotocolbasedonsimultaneousmeasurementoftheseobservables
x y z
was first considered in Ref.[16].) In contrast to the above purification protocols there is no need
to perform Hamiltonian feedback. In this sense it is an “open loop” protocol, similar to the
“random unitary control” purification protocols introduced in Ref. [17] for spin-j systems. Though
operationally different, the two protocols can be shown to be equivalent in the special (j = 1/2)
case, which is another consequence of the isotropy of the qubit evolution discussed below.
3
R (t)
Z
Z-detector
!
r RY(t)
Bloch Y-detector
R (t)
X
X-detector
FIG. 1: A qubit measured by three orthogonal detectors.
Considering the rate of average purification under ideal measurements (with detector efficiency
η = 1) the purification speed-up is 4, (this speed-up is implicit in the results of Ref. [15]). Here
the comparison is with a single measurement of the same strength with no feedback (the speed-up
of 4/3 quoted in Ref. [17] comes from keeping the total measurement strength the same in the
two protocols). However, we show here that by the alternate metric — the mean time T of first
h i
passage — the speed-up is only 2. Both of these purification speed-ups can be understood via
the isotropy of the qubit evolution [15] in the Bloch space that allows one to represent the three
detectors by another equivalent detector triad, σˆ , σˆ , σˆ , measuring at each time moment along
x′ y′ z′
the state and in the complementary directions.
We also study the purification dynamics for inefficient detectors, η = 1 δ < 1. In the
−
asymptotic high-purity limit (ε,δ 1) the increase ∆T¯(δ,ε) of the mean time of first passage T
≪ h i
shows a scaling behavior (i.e. it dependsonly on the ratio δ/ε) as seen from numerical calculations.
The increase ∆T¯(δ/ε) is linear for relatively small inefficiency but grows exponentially, reaching
exp(δ/2ε) for relatively large inefficiency.
∼
II. FURTHER BACKGROUND: SINGLE DETECTOR PURIFICATION PROTOCOLS
A. Qubit impurity from single-detector measurement
For a quantum-limited detector, in the limit of infinite detector bandwidth, the evolution of
the state ρ(t) of a quantum system due to weak continuous measurements of a variable X can be
described by the stochastic master equation (SME) [2]:
Γ Γ
0 0
dρ = dt [X]ρ+ dW(t) [X]ρ. (1)
2 D r 2 H
4
Here [A]ρ AρA† 1(A†Aρ + ρA†A), [A]ρ Aρ + ρA† Tr[(A† + A)ρ]ρ, and dW is the
D ≡ − 2 H ≡ −
increment of a Wiener noise process. The internal Hamiltonian evolution of the system is either
neglected (in case of a large measurement strength Γ ), or eliminated (see, e.g. Ref. [11]). The
0
measurement rate Γ sets the rate at which information (about the final state) is extracted, and
0
thus the rate at which the system is projected onto a single eigenstate of X [18, 19] (provided the
spectrum of X is non-degenerate). We will specialize on the measurement of a qubit and choose
first X = σˆ , although some results below are valid for a general X.
z
The measurement record in a small time interval [t,t+dt) is given by
dR(t) = X dt+dW(t)/ 2Γ , (2)
0
h i
p
where X = Tr[Xρ(t)], and dW(t) is the same realization of the Wiener increment that appears
h i
in Eq. (1), thus providing explicit dependence of the state evolution on the specific measurement
result. The evolution generated via Eq. (1) corresponds to infinitesimal measurement operators
(see, e.g. Refs. [2, 11, 23]). Since the same observable X is measured at each time step, these
operators commute, allowing one to integrate Eq. (1) exactly. Thus, for a finite time interval [0,τ)
the final state will depend only on the time-averaged measurement record:
1 τ
µ(τ) = dt′dR(t′). (3)
τ Z
0
Introducing the measurement operator [2]
1/4
Γ τ Γ τ
M 0 exp[ (µ σˆ )2 0 ]
µ,z z
≡ (cid:18) π (cid:19) − − 2
one obtains, using the method of un-normalized density matrices (see, e.g. Refs. [20–23]):
†
M ρ(0)M
µ,z µ,z
ρ(τ,µ) = . (4)
P(µ)
†
Here the probability distribution for µ(τ) is given by P(µ) = Tr[ρM M ].
µ,z µ,z
In the basis of the observable X, i , the above result can be written as a quantum
{| i}
Bayesian filter [18, 24]: the update of the diagonal density matrix elements ρ will look ex-
ii
actly as a classical Bayesian update of a “probability distribution”: ρ (τ) = ρ (0)P(µ i)/P(µ),
ii ii
|
with Gaussian likelihoods P(µ i); this was termed in Ref. [18] the quantum-classical correspon-
|
dence principle. The outcomes µ will be Gaussian distributed around the eigenvalues of X, x :
i
{ }
P(µ i) = Γ τ/πexp[ (µ x )2Γ τ]withvarianceVar = 1/2Γ τ,andexpectedoutcomeprobabil-
0 i 0 0
| − −
p
ity P(µ) = ρ P(µ i). The update of the non-diagonal matrix elements in (4) will be according
i ii |
P
to the rule: ρ (τ)= ρ (0) ρ (τ)ρ (τ)/ρ (0)ρ (0) provided the measurement is preformed by a
ij ij ii jj ii jj
p
5
quantum-limited detector; this implies that a pure state will remain pure [2]. From the above it is
clear that the measurement time τ to distinguish approximately two eigenvalues, x ,x is set
meas i j
by Var = (x x )2/4. For a qubit this gives τ = 1/2Γ [18].
|τmeas i− j meas 0
For further use we also write down the quantum state evolution in terms of the Bloch vector
components of thestate, x = 2Reρ , y = 2Imρ , z = ρ ρ . From Eq.(1)forthemeasurement
12 12 11 22
−
of σˆ it follows:
z
dz = (1 z2) 2Γ dW
0
−
Γ p
0
dx = x zx 2Γ dW
0
− η −
p
Γ
0
dy = y zy 2Γ dW. (5)
0
− η −
p
In these equations we also included the effect of detector non-ideality (inefficiency) η, leading to a
pure dephasing : In the ensemble averaged equations (since Eqs.(5) are in the Itˆo form, averaging
means just to drop the noise term, and corresponds to ignoring the detector results), Γ is the
0
decoherence rate due to an ideal detector of measurement rate Γ , and η = Γ /(Γ +γ) 1 is the
0 0 0
≤
detector efficiency (ideality), defined as the ratio of Γ to the total decoherence Γ = Γ +γ. For
0 0
a single detector the simplest model to describe the extra decoherence γ is to consider a second
independent detector “in parallel” (i.e., measuring the same σˆ variable) by adding terms similar
z
to Eq. (1) with Γ replaced by γ, and subsequently averaging over that detector output. Thus,
0
the density matrix available for an observer who takes into account only the results of the first
detector will be described by Eqs.(5).
B. Single-detector purification protocols
Weconsiderfirstpurificationprotocolsviasingle-detectormeasurementwithanideal(quantum-
limited) detector.
1. Purification without feedback
For a single detector measurement without feedback, the state evolution in the detector basis is
essentiallyclassical, Eq.(4). Thatis,ifthedensitymatrixbeginsdiagonalinthemeasurementbasis
(as it will be if it is a completely mixed state), it remains so and there is no way to distinguish
ρ (t),ρ (t) from a classical probability distribution [18, 25]. For continuous measurement
11 22
{ }
one can consider the purity or entropy of the monitored state, and in this particular case the
6
von Neumann entropy of the state coincides with the Shannon entropy (see, e.g. Refs. [1, 28]):
S = Tr[ρˆlnρˆ] = (ρ lnρ +ρ lnρ ). In what follows we will consider the so called linear
vN 11 11 22 22
− −
entropy (see, e.g. [26]), s = 1 Trρˆ2 1 p which is a monotonic function of S . Here
vN
− ≡ −
p Trρˆ2 = (1+x2+y2+z2) is the purity expressed through the Bloch components of the state.
≡ 2
The corresponding equation for the purity (at η = 1) follows from Eq. (5):
dp = 2Γ (1 p)(1 z2) dt+2 2Γ z(1 p)dW. (6)
0 0
− − −
(cid:2) (cid:3) p
Since purity p, like entropy, is invariant under unitary transformations, without loss of generality
onecansaythatmeasurement“parallelto”(inthesamebasisas)thestatecorrespondstox = y = 0
and
dp =Γ [1 z2]2dt+ 2Γ z[1 z2]dW. (7)
0 0
− −
p
so that the average change of purity is dp = Γ (1 z2)2dt. We note that the same result holds
k 0
h i −
for non-ideal measurement, η < 1, since the detector’s non-ideality affects only the evolution of the
non-diagonal density matrix elements; this means that a non-ideal detector will purify the state
if the measurement is along the state. It is clear that Γ plays the role of a maximal classical
0
purification (information acquisition) rate that happen when z = ρ ρ = 0, i.e. when the two
11 22
−
outcomes are equally likely (see, e.g., Ref. [28]). By approaching z 1, dp 0 since “little
k
→ ± h i →
information” remains to be extracted to clarify that collapse has happened [29].
2. Jacobs feedback purification protocol
From the discussion above, it is intuitively clear how the Jacobs’ enhanced purification protocol
works. Giventhestateρ(t), oneshouldcontinuouslyadaptthemeasurementbasis[or,equivalently,
rotatethestatetosomeρ′(t)]sothatthedetector wouldperformmeasurementinacomplementary
direction with respect to the eigenbasis of the rotated state (i.e., measuring perpendicular to the
state, in the Bloch picture). In the detector basis the rotated state again possesses equally likely
outcomes, with ρ′ (t) = ρ′ (t). As expected classically [28], this procedure maximizes the average
11 22
purification rate since p = p′ under rotation, while z z′ = 0 in Eq. (6). However, the possibility
→
to perform coherent rotations of the density matrix is of course a quantum mechanical effect. The
procedure also makes the purity evolution deterministic [8], i.e. the noise term in Eq. (6) is zeroed
so that
dp = 2Γ (1 p)dt (8)
0
−
7
and p = p under this protocol. From this one can evaluate the time τ when the average purity
⊥
h i
p reaches a given level 1 ε:
h i −
1 1−ε dp 1
τ lnε−1. (9)
⊥
≃ 2Γ Z 1 p ≃ 2Γ
0 0
−
It should be noted that an attempt to evaluate the analogous time τ in the case of the classical
k
measurement along the state, using dp from Eq. (7), will lead to a wrong scaling of 1/ε.
k
h i ∼
The reason is that dp = d p in general, and the evolving purity distribution (p,t) is generally
h i 6 h i P
different from δ-function [10]. The correct evaluation of τ is to calculate the average purity p (t)
k
h i
by taking into account the exact solution, Eq. (4), of the stochastic evolution equations (1). In the
high-purity limit this leads to [8–11]
1
τ lnε−1 (10)
k
≃ Γ
0
This is exactly twice as long as the time in Eq. (9), which establishes the speed-up of 2 for the
Jacobsprotocol. Wenotehowever, that, unlikethecaseofparallelmeasurement, theperpendicular
measurement will not purify to a completely pure state if the measurement is inefficient, as will
be explored in Sec. IIIC. The reason is that the excess back-action in an inefficient perpendicular
measurements results in decay in the coherences of the state.
3. Wiseman and Ralph purification protocol
Instead of considering the time τ at which the average purity p reaches a certain level 1 ε,
h i −
one can also consider the average time T for a system to attain that purity level, p(T) = 1 ε
h i −
[10, 12, 13]. It was noted in Ref. [10] that in many circumstances the time T is a more useful
h i
quantity. The reason is that the time T at which p(T) = 1 ε, has a well-behaved statistics [10],
−
in contrast with p(t) which has extremely long tails at relatively small values of p. That is, the
averaged p is strongly influenced by the rare cases that are slow to purify. Because of this there
h i
is a substantial disagreement between τ and T for a qubit. It was shown in Ref. [10], however,
h i
that good agreement is found between T and Tlog, defined as the time required for ln[1 p(t)]
h i h − i
to reach the certain level lnε. This is because taking the logarithm de-emphasizes the tails, and
indeed for a qubit ln(1 p) has near-normal distribution [10].
−
Therefore, we consider the stochastic equation for the logarithm of the linear entropy s 1 p.
≡ −
It follows from Eq. (6) that:
dlns = 2Γ 2s+x2+2z2 dt+2 2Γ zdW (11)
0 0
− { }
p
8
(Here, without loss of generality we have put y = 0, since a single σˆ -measurement keeps the
z
state of the qubit in a fixed meridional plane). In the Wiseman-Ralph feedback protocol [10] one
keeps the monitored state along the detector z-axis (so x2 = 0, while z2 is maximized at 1 2s).
−
Thus, the single detector purification is maximized and in the high-purity limit, s 0, we obtain
≈
hdlnsik 4Γ . Using this, the average time (of first passage) T is evaluated as T Tlog and
dt ≃ − 0 h i h i ≃
in the high-purity limit, ε 1
≪
1
Tlog = lnε−1, (12)
k 4Γ
0
which is half the size of τ , and 4 times shorter than τ . Inwhat follows we will usethese results to
⊥ k
understandthepurificationviathreesimultaneouscomplementarymeasurementswithoutfeedback.
III. CONTINUOUS MEASUREMENT OF THREE COMPLEMENTARY QUBIT
VARIABLES
We considercontinuous measurementof thequbitcomplementary observables σˆ , σˆ , andσˆ by
x y z
three independent linear detectors (see Fig. 1). In principle, given a σˆ -detector, the measurement
z
of, say, σˆ can be implemented operationally by performing fast unitary rotation, U (t), of the
x x
state towards the x-axis “at the beginning” of the measurement interval dt, then a continuous
measurement with the z-detector and finally, a backwards transformation, U−1(t + dt) “at the
x
end” , in any infinitesimal measurement time step dt, as discussed in Ref. [17]. We also mention
that a simultaneous measurement of the three observables is possible to implement, at least in
principle, in a quantum optics setup [27] (e.g., using a single atom in an optical cavity field).
In the case of detectors measuring the qubit in the bases of non-commuting observables, it is
impossibletousethequantumBayesrule(4)forfinitetimesbecausethemeasurementback-actions,
M , M , M , do not commute with each other. The measurement back-actions commute
µ,x µ,y µ,z
only for small measurement time intervals dt so that one should apply the POVM update in its
differential form Eq. (1) and then sum up the contributions to the qubit evolution. For the case of
mutually unbiased observables σˆ , σˆ , and σˆ it is convenient to use the Bloch vector components,
x y z
r = (x,y,z) = Tr[ρˆσˆ ],k = x,y,z determined by the time-dependent qubit density matrix ρˆ.
k
{ }
From Eq. (5) for the influence due to σˆ -measurement, the influences due to σˆ (σˆ )-measurement
z x y
can be obtained by cyclic replacements z x y, bearing in mind that the variables x(y) should
→ →
playthesameroleaszinaσˆ -measurement. Inthiswayweobtainthefollowingevolutionequations
z
in Itˆo form (see Ref. [15]) that take into account the three measurement records dR (t),k = x,y,z
k
9
asinEq.(2),andcorrespondinglyintroducethreeindependentWienerprocesses,dW dW = δ dt:
i k ik
dx = (Γ +Γ )x+(1 x2) 2Γ dW
y z 0,x x
− −
p
xy 2Γ dW xz 2Γ dW (13)
0,y y 0,z z
− −
p p
dy = (Γ +Γ )y+(1 y2) 2Γ dW
z x 0,y y
− −
p
yz 2Γ dW xy 2Γ dW (14)
0,z z 0,x x
− −
p p
dz = (Γ +Γ )z+(1 z2) 2Γ dW
x y 0,z z
− −
p
xz 2Γ dW yz 2Γ dW (15)
0,x x 0,y y
− −
p p
Here Γ γ +Γ are the total decoherence rates for each detector, including individual dephas-
k k 0,k
≡
ings γ and measurement rates Γ ,k = x,y,z. Note that a pure dephasing, say in the x-basis,
k 0,k
causes the decay of the z- and y-components. Eqs. (14) and (15) are simply obtained by cyclic
permutation of the variables in Eq. (13).
A. Qubit evolution with identical detectors
In what follows we consider the simplest (but still rich) case of three identical detectors: Γ =
0,k
Γ and γ = γ 0. Then the qubit evolution (13)–(15) can be rewritten in a vector form as
0 k
≥
dr = 2Γrdt+ 2Γ dW(t)(1 r2) [r [r dW(t)]] , (16)
0
− − − × ×
p (cid:8) (cid:9)
where dW(t) dW ,dW ,dW is the vector of Wiener increments corresponding to the vector
x y z
≡ { }
ofresultsdR = dR ,dR ,dR ,similartoEq.(2). Theevolution (16)isinvariantunderarbitrary
x y z
{ }
rotations (see Fig. 2) of the coordinate system in Bloch space [30] (as is the ensemble averaged
evolution: r˙ = 2Γr). Hence while measurement of only a single observable σˆ “attracts” the
k
−
qubit state to one of the corresponding eigenvectors, the simultaneous measurement of σˆ , σˆ ,
x y
and σˆ leads to no preferable direction in the Bloch space. In Ref. [15] this isotropy of the qubit
z
evolution was shown to lead to a locally isotropic Brownian diffusion of the direction of the Bloch
vector. In particular, for pure states under ideal measurements, the point on the Bloch sphere
diffuses isotropically [31] with a diffusion coefficient Γ [15]. The evolution isotropy was used then
0
toconstructsimplequantumstate estimations [15]. Inwhatfollows, wewillusethequbitevolution
isotropy to understand the qubit purification dynamics.
10
z’ ! !
z
dR ! dW
y
x’
!
r
y’
x
Bloch
FIG. 2: Bloch equations invariance under rotations for three complementary measurements with identical
detectors. The evolution is determined by the relative orientation of the Bloch vector and the vector of
results [15].
B. Purification dynamics with three complementary measurements
Eq.(16)showsthatthemeasurementtermcomponentalongtheradiusis√2Γ0(dW er)(1 r2)
· −
and vanishes on the sphere. The decreasing r-dependent coefficient suggests that r = 1 is an
attractor of the random evolution. Thus, the qubit state will purify in the ideal case (γ = 0) for
any realization of the measurement process.
To establish further the purification dynamics, we start with Eq. (16) and transform it to polar
coordinates, r, θ, φ. The equation for the radius decouples from the other two and reads [15]:
dr = 2Γ (1/r r/η)dt+ 2Γ (1 r2)dW , (17)
0 0 r
− −
p
where dWr dW er is the projection of the noise vector onto the Bloch vector direction,
≡ ·
er = (sinθcosϕ,sinθsinϕ,cosθ), and the normalization of the noise is dWrdWr = dt.
AsseenfromEq.(17),ifr < η1/2,thenonaverager˙ > 0,whichmeansstatepurificationforideal
measurements, η = 1. Since the equation for r is singular at the origin, it will be more convenient
to consider further the dynamics of the purity p = Trρˆ2 = (1+r2). In this choice 1/2 < p < 1, i.e.,
2
a totally mixed state corresponds to p = 1/2. We find
dp = 2Γ 1 (2p 1)/η+2(1 p)2 dt+2 2Γ (1 p) 2p 1dW . (18)
0 0 r
{ − − − } − −
p p
For an ideal measurement, η = 1, and starting from a non-pure initial state, the qubit will purify
(p 1) on a time scale of the order of τ Γ−1. For a non-ideal measurement, η < 1, purity
→ meas ∼ 0
will continue to fluctuate around a stationary average value, p < 1.
st
h i